Armature deform on same placed verticies of different objects creates weird gaps
If two meshes have no unapplied scale/rotation/location, any vertices in the exact same locations, with the exact same weights, will be deformed in the exact same way by two identical armature modifiers.  (Same armature, same deformation mode-- "preserve volume".)

To make sure that vertices are in the exact same location, you can always shrinkwrap to the nearest vertex and apply.  To make sure that vertices have the exact same weights, you can data transfer from nearest vertex and apply.

To see the specific weights of a vertex, select that vertex in edit or weight paint mode and look on the sidebar, item tab, vertex groups section.

To make sure that meshes are using the exact same armature modifier, you can copy modifiers from the link menu (ctrl-L for me.)

Note that modifiers evaluated before the armature can change the position and/or weights of verts.  What matters are the values at the time that the armature modifier is evaluated.

As an aside specific to this picture, I've found that a surface deform is a much easier way to deform mouth interiors.  Because of differences in topology, you can get clipping (not from the margin verts, not if you're careful about weights and positions) from just an armature modifier.